Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.1
Options on
Stock Indices,Currencies,and
Futures
Chapter 12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.2European Options on Stocks
Paying Continuous Dividends
We get the same probability distribution for
the stock price at time T in each of the
following cases
1,The stock starts at price S0 and provides a
continuous dividend yield = q
2,The stock starts at price S0e-qT and provides
no dividend yield
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.3European Options on Stocks
Paying Continuous Dividends
(continued)
We can value European options by reducing
the stock price to S0e–q T and then behaving
as though there is NO dividend
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.4
Extension of Chapter 7 Results
(Equations 12.1 to 12.3)
0
qT rTcS e Xe
Lower Bound for calls:
Lower Bound for puts
0
rT qTp X e S e
Put Call Parity
0r TT qc X e p eS
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.5
Extension of Chapter 11
Results (Equations 12.4 and 12.5)
0 1 2
2 0 1
0
1
0
2
( ) ( )
( ) ( )
2
l n( / ) ( / 2)
w he r e
2
l n( / ) ( / 2)
rq
r
T
T
T
T q
c S N d Xe N d
p Xe N d S N d
S X r T
d
T
S X r T
d
e
e
q
q
T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.6
The Binomial Model
S0u
u
S0d
d
S0
f=e-rT[pfu+(1-p)fd ]
p=?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.7
The Binomial Model
(continued)
In a risk-neutral world the stock price grows
at r-q rather than at r when there is a
dividend yield at rate q
The probability,p,of an up movement must
therefore satisfy
pS0u+(1-p)S0d=S0e(r-q)T
so that
()rTqed
p
ud
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.8
Index Options
Option contracts are on 100× the index
The most popular underlying indices are
the S&P 100 (American) OEX
the S&P 500 (European) SPX
the Major Market Index (XMI)
Contracts are settled in cash
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.9
Index Option Example
Consider a call option on the OEX index
with a strike price of 560
Suppose 1 contract is exercised
when the index level is 580
What is the payoff?
(Ans=2000)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.10
Valuing European Index
Options
We can use the formula for an option on a
stock paying a continuous dividend yield
Set S0 = current index level
Set q = average dividend yield expected
during the life of the option
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.11
Currency Options
Currency options trade on the Philadelphia
Exchange (PHLX)
They are used by corporations to hedge their
FX exposure
The size of 1 contract depends on the currency
[see Table 13.3 (p,283)]
There also exists an active over-the-counter
(OTC) market
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.12
The Foreign Interest Rate
We denote the foreign interest rate by rf
When a U.S,company buys one unit of the
foreign currency it has an investment of S0
dollars
The return from investing at the foreign
rate is rf S0 dollars
This shows that the foreign currency
provides a,dividend yield” at rate rf
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.13
Valuing European
Currency Options
A foreign currency is an instrument that provides a
continuous income (“dividend yield” )
= foreign risk-free rate (rf)
We can use the formula for an option on a stock
paying a continuous dividend yield:
– Set S0 = current exchange rate (domestic/foreign)
– Set q = rf
Note that the income earned in the
domestic currency is rf S0
showing that rf is analogous to q
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.14
European Options on
Currency Options
0 1 2
2 0 1
e ( ) e ( )
e ( ) e ( )
f
f
-T - r T
-T-r
r
rT
c S N d X N d
p X N d S N d
2
0
1
2
0
21
l n ( / ) ( / 2 )
l n ( / ) ( / 2 )
f
f
rS X r T
d
T
S X r T
d d T
r
T
where
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.15
Alternative Formulas
(Equations 12.11 and 12.12,page 284)
()
00 f
rTrF S eUsing
c e F N d XN d
p e XN d F N d
d
F X T
T
d d T
rT
rT
[ ( ) ( )]
[ ( ) ( )]
ln( / ) /
0 1 2
2 0 1
1
0
2
2 1
2?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.16
Futures Options
When a CALL is exercised the holder acquires a
LONG futures position + cash equal to the excess of
the futures price over the strike price
When a PUT is exercised the holder acquires a
SHORT futures position + cash equal to the excess of
the strike price over the futures price
Contract size on S&P500 is 250 times the index
(changed from 500 on Oct,31,1997)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.17
The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F0-X
Payoff from put = X-F0
where F0 is futures price at time of exercise
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.18
Type of Futures Options
(Table 12.4)
Options on
-- Agricultural futures
-- Oil futures
-- Livestock futures
-- Interest rate futures
-- Currency futures
-- Index futures
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.19Reasons for the Popularity of
Futures Options
(compared with spot options)
Cheaper and more convenient to deliver futures
contracts on the assets rather than the asset itself
Normally settled in cash,It is appealing to those
investors with limited capital
Trading of futures and futures options are arranged in
pits side by side in the exchange,This facilitates
hedging,arbitrage and speculation
Lower transaction costs
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.20
Growth Rates For Futures Prices
A futures contract requires NO initial investment
In a risk-neutral world the expected return
should be 0
The expected growth rate of the futures price
is therefore 0
The futures price can therefore be treated like a
stock paying a dividend yield of r
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.21
Valuing European Futures Options
We can use the formula for an option on a stock
which pays a continuous dividend yield:
–Set S0 = current futures price (F0)
–Set q = domestic risk-free rate (r)
Setting q = r ensures that the expected growth
of F0 in a risk-neutral world is ZERO
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.22
Black’s Formula
The formulas for European options on futures are
sometimes referred to as Black’s formulas
0 1 2
2 0 1
e [ ( ) ( ) ]
e [ ( ) ( ) ]
- r T
- r T
c F N d X N d
p X N d F N d
2
0
1
2
0
21
l n( / ) ( / 2)
l n( / ) ( / 2)
F X T
d
T
F X T
d d T
T
where
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.23
Put-Call Parity for Futures
Options (Equation 12.13,page 291)
Consider the following two portfolios:
1,European call plus Xe-rT of cash
2,European put plus long futures plus
cash equal to F0e-rT
They are worth the same at time T and
thus worth the same today,so that
c+Xe-rT=p+F0 e-rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.24
Put-Call Parity (summary)
Indices
Foreign Exchange
Futures
0eer T q Tc X p S
0ee f
rTrTc X p S
0eer T r Tc X p F
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.25
Summary of Key Results
We can treat stock indices,currencies,and
futures like a stock paying a
continuous dividend yield of q
– For stock indices,q = average dividend yield
on the index over the option life
– For currencies,q = rf
– For futures,q = r
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.26
Futures Price = $33
Option Price = $4
Futures Price = $28
Option Price = $0
Futures price = $30
Option Price=?
Binomial Tree Example
A 1-month call option on futures has a strike price of 29,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.27
Consider the Portfolio,long D futures
short 1 call option
Portfolio is riskless when 3D – 4 = -2D or
D = 0.8
3D – 4
-2D
Setting Up a Riskless Portfolio
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.28
Valuing the Portfolio
( Risk-Free Rate is 6% )
The riskless portfolio is,
long 0.8 futures
short 1 call option
The value of the portfolio in 1 month is
3x0.8-4 = -1.6
The value of the portfolio today is
-1.6e – 0.06/12 = -1.592
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.29
Valuing the Option
The portfolio today that is
long 0.8 futures
short 1 option
is worth -1.592
The value of the futures today is zero
The value of the option must therefore
be 1.592
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.30
Generalization of Binomial
Tree Example (Figure 12.3,page 292)
A derivative lasts for time T & is
dependent on a futures
F0u
u
F0d
d
F0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.31
Generalization
(continued)
Consider the portfolio that is long D futures and
short 1 derivative
The portfolio is riskless when
D
u df
F u F d0 0
F0u D? F0 D –?u
F0d D? F0D –?d
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.32
Generalization
(continued)
Value of the portfolio at time T is
F0u D–F0D–?u
Value of portfolio today is
–?
Hence
= – [F0u D –F0D –?u]e-rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.33
Generalization
(continued)
Substituting for D we obtain
= [ p?u + (1 – p )?d ]e–rT
where
In general (eq.12.7 on p277)
p du d1
()rqed
p ud
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.34
Futures Option Prices vs,Spot
Option Prices
If futures prices are higher than spot prices
(normal market),an American call on futures
is worth more than a similar American call on
spot,An American put on futures is worth less
than a similar American put on spot
When futures prices are lower than spot
prices (inverted market) the reverse is true
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.35Using Index Options for
Portfolio Insurance
Suppose the value of the index is S0 and the
strike price is X
If a portfolio has a? of 1.0,the portfolio manager
BUYS 1 put option for each 100S0 dollars held
If the? is not 1.0,the portfolio manager BUYS
put option for each 100S0 dollars held
In both cases,X is chosen to give the appropriate
insurance level
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.36
Example 1 of Index Options for
Portfolio Insurance
Portfolio has a?=1.0
It is currently worth $5 million
The index currently stands at 1000
What trade is necessary to provide insurance
against the portfolio value falling below $4.8
million?
(Ans,buy 50 put option with strike price of 960)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.37
Example 2 of Index Options for
Portfolio Insurance
You are worried about a market drop,You have a $1 million
portfolio that has done very well in the recent rising market
because it has a?=2,Suppose further that the current risk
free rate is 12% pa and the dividend rate on the market index
and your portfolio is 4% pa,You want to ensure that your
portfolio does not drop below 90% of its current value and the
S&P 100 index is at 250,What do you do?
First you need to calculate the value of your portfolio for a
variety of market returns,To do that,let’s first review the
CAPM,])([)( rrErrE
mpp
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.38
Example 2 for Portfolio Insurance
(continued)
So,lets look at the value of your portfolio if the
index increases to 260
Value of the Index in 3 months 260
Return from change in index (260-250)/250 = 4% per 3 mths
Dividends from Index 0.25 * 4% = 1% per 3 mths
Total Index Return 4% + 1% = 5% per 3 mths
Risk-free rate 0.25 * 12% = 3% per 3 mths
Excess Index Return 5% - 3% = 2% per 3 mths
Excess Portfolio Return 2% * 2 = 4% per 3 mths
Portfolio Return 4% + 3% = 7% per 3 mths
Dividends from Portfolio 0.25 * 4% = 1% per 3 mths
Increase in Value of Portfolio 7% - 1% = 6%
Value of Portfolio $1mill * (1 + 6%) = $1.06 mill
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.39
Example 2 for Portfolio Insurance
(continued)
The following table illustrates the value of the
portfolio for various levels of the index
Value of Value of Portfolio
Index in in Three Months
Three Months (millions of dollars)
270 1.14
260 1.06
250 0.98
240 0.90
230 0.82
(In reality,you do not need to do
this trial and error process,you
can just reverse the procedure on
the previous slide.)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.40
Example 2 for Portfolio Insurance
(continued)
So,lets reverse the process and look at the value
of the index if your portfolio decreases 10%
Value of Portfolio $0.90 million
Increase in Value of Portfolio -10%
Dividends from Portfolio 0.25 * 4% = 1% per 3 mths
Portfolio Return -10% + 1% = -9% per 3 mths
Risk-free rate 0.25 * 12% = 3% per 3 mths
Excess Portfolio Return -9% - 3% = -12% per 3 mths
Excess Index Return -12% / 2 = -6% per 3 mths
Total Index Return -6% + 3% = -3% per 3 mths
Dividends from Index 0.25 * 4% = 1% per 3 mths
Return from Change in index -3% - 1%= -4% per 3 mths
Value of the Index in 3 months (1 - 4%)* 250 = 240
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.41
Example 2 for Portfolio Insurance
(continued)
Expanding the CAPM
%404.0)(01.003.0
)(03.006.0
)(2/12.0
)(/)03.009.0(
])([01.010.0
])([)10.0(
])([)(
])([)(
mc
mcmd
mdmc
mdmcP
mdmcP
mdmcPpd
mdmcPpdpc
mpp
rE
rEr
rrEr
rrrE
rrrEr
rrrErrE
rrrErrrE
rrErrE
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.42
Example 2 for Portfolio Insurance
(continued)
From the table it is easy to see that when the
index drops to 240 the value of the portfolio drops
to $0.9 million,Hence,we want to buy puts with a
strike of 240,But,how many puts do we buy?
Each put is worth 100S = 100 * $250 = $25,000,
So for our $1 million portfolio we will need
p u t s8040*0.20 0 0,25 0 0 0,0 0 0,1*
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.43
Assignments
12.1 12.4 12.7 12.9 12.13 12.18 12.19
12.24+12.25 12.31+12.32 12.33
Assignments Questions
Tang Yincai,? 2003,Shanghai Normal University
12.1
Options on
Stock Indices,Currencies,and
Futures
Chapter 12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.2European Options on Stocks
Paying Continuous Dividends
We get the same probability distribution for
the stock price at time T in each of the
following cases
1,The stock starts at price S0 and provides a
continuous dividend yield = q
2,The stock starts at price S0e-qT and provides
no dividend yield
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.3European Options on Stocks
Paying Continuous Dividends
(continued)
We can value European options by reducing
the stock price to S0e–q T and then behaving
as though there is NO dividend
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.4
Extension of Chapter 7 Results
(Equations 12.1 to 12.3)
0
qT rTcS e Xe
Lower Bound for calls:
Lower Bound for puts
0
rT qTp X e S e
Put Call Parity
0r TT qc X e p eS
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.5
Extension of Chapter 11
Results (Equations 12.4 and 12.5)
0 1 2
2 0 1
0
1
0
2
( ) ( )
( ) ( )
2
l n( / ) ( / 2)
w he r e
2
l n( / ) ( / 2)
rq
r
T
T
T
T q
c S N d Xe N d
p Xe N d S N d
S X r T
d
T
S X r T
d
e
e
q
q
T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.6
The Binomial Model
S0u
u
S0d
d
S0
f=e-rT[pfu+(1-p)fd ]
p=?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.7
The Binomial Model
(continued)
In a risk-neutral world the stock price grows
at r-q rather than at r when there is a
dividend yield at rate q
The probability,p,of an up movement must
therefore satisfy
pS0u+(1-p)S0d=S0e(r-q)T
so that
()rTqed
p
ud
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.8
Index Options
Option contracts are on 100× the index
The most popular underlying indices are
the S&P 100 (American) OEX
the S&P 500 (European) SPX
the Major Market Index (XMI)
Contracts are settled in cash
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.9
Index Option Example
Consider a call option on the OEX index
with a strike price of 560
Suppose 1 contract is exercised
when the index level is 580
What is the payoff?
(Ans=2000)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.10
Valuing European Index
Options
We can use the formula for an option on a
stock paying a continuous dividend yield
Set S0 = current index level
Set q = average dividend yield expected
during the life of the option
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.11
Currency Options
Currency options trade on the Philadelphia
Exchange (PHLX)
They are used by corporations to hedge their
FX exposure
The size of 1 contract depends on the currency
[see Table 13.3 (p,283)]
There also exists an active over-the-counter
(OTC) market
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.12
The Foreign Interest Rate
We denote the foreign interest rate by rf
When a U.S,company buys one unit of the
foreign currency it has an investment of S0
dollars
The return from investing at the foreign
rate is rf S0 dollars
This shows that the foreign currency
provides a,dividend yield” at rate rf
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.13
Valuing European
Currency Options
A foreign currency is an instrument that provides a
continuous income (“dividend yield” )
= foreign risk-free rate (rf)
We can use the formula for an option on a stock
paying a continuous dividend yield:
– Set S0 = current exchange rate (domestic/foreign)
– Set q = rf
Note that the income earned in the
domestic currency is rf S0
showing that rf is analogous to q
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.14
European Options on
Currency Options
0 1 2
2 0 1
e ( ) e ( )
e ( ) e ( )
f
f
-T - r T
-T-r
r
rT
c S N d X N d
p X N d S N d
2
0
1
2
0
21
l n ( / ) ( / 2 )
l n ( / ) ( / 2 )
f
f
rS X r T
d
T
S X r T
d d T
r
T
where
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.15
Alternative Formulas
(Equations 12.11 and 12.12,page 284)
()
00 f
rTrF S eUsing
c e F N d XN d
p e XN d F N d
d
F X T
T
d d T
rT
rT
[ ( ) ( )]
[ ( ) ( )]
ln( / ) /
0 1 2
2 0 1
1
0
2
2 1
2?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.16
Futures Options
When a CALL is exercised the holder acquires a
LONG futures position + cash equal to the excess of
the futures price over the strike price
When a PUT is exercised the holder acquires a
SHORT futures position + cash equal to the excess of
the strike price over the futures price
Contract size on S&P500 is 250 times the index
(changed from 500 on Oct,31,1997)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.17
The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F0-X
Payoff from put = X-F0
where F0 is futures price at time of exercise
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.18
Type of Futures Options
(Table 12.4)
Options on
-- Agricultural futures
-- Oil futures
-- Livestock futures
-- Interest rate futures
-- Currency futures
-- Index futures
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.19Reasons for the Popularity of
Futures Options
(compared with spot options)
Cheaper and more convenient to deliver futures
contracts on the assets rather than the asset itself
Normally settled in cash,It is appealing to those
investors with limited capital
Trading of futures and futures options are arranged in
pits side by side in the exchange,This facilitates
hedging,arbitrage and speculation
Lower transaction costs
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.20
Growth Rates For Futures Prices
A futures contract requires NO initial investment
In a risk-neutral world the expected return
should be 0
The expected growth rate of the futures price
is therefore 0
The futures price can therefore be treated like a
stock paying a dividend yield of r
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.21
Valuing European Futures Options
We can use the formula for an option on a stock
which pays a continuous dividend yield:
–Set S0 = current futures price (F0)
–Set q = domestic risk-free rate (r)
Setting q = r ensures that the expected growth
of F0 in a risk-neutral world is ZERO
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.22
Black’s Formula
The formulas for European options on futures are
sometimes referred to as Black’s formulas
0 1 2
2 0 1
e [ ( ) ( ) ]
e [ ( ) ( ) ]
- r T
- r T
c F N d X N d
p X N d F N d
2
0
1
2
0
21
l n( / ) ( / 2)
l n( / ) ( / 2)
F X T
d
T
F X T
d d T
T
where
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.23
Put-Call Parity for Futures
Options (Equation 12.13,page 291)
Consider the following two portfolios:
1,European call plus Xe-rT of cash
2,European put plus long futures plus
cash equal to F0e-rT
They are worth the same at time T and
thus worth the same today,so that
c+Xe-rT=p+F0 e-rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.24
Put-Call Parity (summary)
Indices
Foreign Exchange
Futures
0eer T q Tc X p S
0ee f
rTrTc X p S
0eer T r Tc X p F
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.25
Summary of Key Results
We can treat stock indices,currencies,and
futures like a stock paying a
continuous dividend yield of q
– For stock indices,q = average dividend yield
on the index over the option life
– For currencies,q = rf
– For futures,q = r
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.26
Futures Price = $33
Option Price = $4
Futures Price = $28
Option Price = $0
Futures price = $30
Option Price=?
Binomial Tree Example
A 1-month call option on futures has a strike price of 29,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.27
Consider the Portfolio,long D futures
short 1 call option
Portfolio is riskless when 3D – 4 = -2D or
D = 0.8
3D – 4
-2D
Setting Up a Riskless Portfolio
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.28
Valuing the Portfolio
( Risk-Free Rate is 6% )
The riskless portfolio is,
long 0.8 futures
short 1 call option
The value of the portfolio in 1 month is
3x0.8-4 = -1.6
The value of the portfolio today is
-1.6e – 0.06/12 = -1.592
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.29
Valuing the Option
The portfolio today that is
long 0.8 futures
short 1 option
is worth -1.592
The value of the futures today is zero
The value of the option must therefore
be 1.592
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.30
Generalization of Binomial
Tree Example (Figure 12.3,page 292)
A derivative lasts for time T & is
dependent on a futures
F0u
u
F0d
d
F0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.31
Generalization
(continued)
Consider the portfolio that is long D futures and
short 1 derivative
The portfolio is riskless when
D
u df
F u F d0 0
F0u D? F0 D –?u
F0d D? F0D –?d
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.32
Generalization
(continued)
Value of the portfolio at time T is
F0u D–F0D–?u
Value of portfolio today is
–?
Hence
= – [F0u D –F0D –?u]e-rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.33
Generalization
(continued)
Substituting for D we obtain
= [ p?u + (1 – p )?d ]e–rT
where
In general (eq.12.7 on p277)
p du d1
()rqed
p ud
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.34
Futures Option Prices vs,Spot
Option Prices
If futures prices are higher than spot prices
(normal market),an American call on futures
is worth more than a similar American call on
spot,An American put on futures is worth less
than a similar American put on spot
When futures prices are lower than spot
prices (inverted market) the reverse is true
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.35Using Index Options for
Portfolio Insurance
Suppose the value of the index is S0 and the
strike price is X
If a portfolio has a? of 1.0,the portfolio manager
BUYS 1 put option for each 100S0 dollars held
If the? is not 1.0,the portfolio manager BUYS
put option for each 100S0 dollars held
In both cases,X is chosen to give the appropriate
insurance level
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.36
Example 1 of Index Options for
Portfolio Insurance
Portfolio has a?=1.0
It is currently worth $5 million
The index currently stands at 1000
What trade is necessary to provide insurance
against the portfolio value falling below $4.8
million?
(Ans,buy 50 put option with strike price of 960)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.37
Example 2 of Index Options for
Portfolio Insurance
You are worried about a market drop,You have a $1 million
portfolio that has done very well in the recent rising market
because it has a?=2,Suppose further that the current risk
free rate is 12% pa and the dividend rate on the market index
and your portfolio is 4% pa,You want to ensure that your
portfolio does not drop below 90% of its current value and the
S&P 100 index is at 250,What do you do?
First you need to calculate the value of your portfolio for a
variety of market returns,To do that,let’s first review the
CAPM,])([)( rrErrE
mpp
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.38
Example 2 for Portfolio Insurance
(continued)
So,lets look at the value of your portfolio if the
index increases to 260
Value of the Index in 3 months 260
Return from change in index (260-250)/250 = 4% per 3 mths
Dividends from Index 0.25 * 4% = 1% per 3 mths
Total Index Return 4% + 1% = 5% per 3 mths
Risk-free rate 0.25 * 12% = 3% per 3 mths
Excess Index Return 5% - 3% = 2% per 3 mths
Excess Portfolio Return 2% * 2 = 4% per 3 mths
Portfolio Return 4% + 3% = 7% per 3 mths
Dividends from Portfolio 0.25 * 4% = 1% per 3 mths
Increase in Value of Portfolio 7% - 1% = 6%
Value of Portfolio $1mill * (1 + 6%) = $1.06 mill
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.39
Example 2 for Portfolio Insurance
(continued)
The following table illustrates the value of the
portfolio for various levels of the index
Value of Value of Portfolio
Index in in Three Months
Three Months (millions of dollars)
270 1.14
260 1.06
250 0.98
240 0.90
230 0.82
(In reality,you do not need to do
this trial and error process,you
can just reverse the procedure on
the previous slide.)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.40
Example 2 for Portfolio Insurance
(continued)
So,lets reverse the process and look at the value
of the index if your portfolio decreases 10%
Value of Portfolio $0.90 million
Increase in Value of Portfolio -10%
Dividends from Portfolio 0.25 * 4% = 1% per 3 mths
Portfolio Return -10% + 1% = -9% per 3 mths
Risk-free rate 0.25 * 12% = 3% per 3 mths
Excess Portfolio Return -9% - 3% = -12% per 3 mths
Excess Index Return -12% / 2 = -6% per 3 mths
Total Index Return -6% + 3% = -3% per 3 mths
Dividends from Index 0.25 * 4% = 1% per 3 mths
Return from Change in index -3% - 1%= -4% per 3 mths
Value of the Index in 3 months (1 - 4%)* 250 = 240
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.41
Example 2 for Portfolio Insurance
(continued)
Expanding the CAPM
%404.0)(01.003.0
)(03.006.0
)(2/12.0
)(/)03.009.0(
])([01.010.0
])([)10.0(
])([)(
])([)(
mc
mcmd
mdmc
mdmcP
mdmcP
mdmcPpd
mdmcPpdpc
mpp
rE
rEr
rrEr
rrrE
rrrEr
rrrErrE
rrrErrrE
rrErrE
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.42
Example 2 for Portfolio Insurance
(continued)
From the table it is easy to see that when the
index drops to 240 the value of the portfolio drops
to $0.9 million,Hence,we want to buy puts with a
strike of 240,But,how many puts do we buy?
Each put is worth 100S = 100 * $250 = $25,000,
So for our $1 million portfolio we will need
p u t s8040*0.20 0 0,25 0 0 0,0 0 0,1*
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
12.43
Assignments
12.1 12.4 12.7 12.9 12.13 12.18 12.19
12.24+12.25 12.31+12.32 12.33
Assignments Questions
